Tony Bernardom

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Capital budgeting in multi-division firms:

Information, agency, and incentives

Antonio E. Bernardo

Hongbin Cai

Jiang Luo

February 24, 2003

We thank Tano Santos and seminar participants at Chicago, Georgia Tech, MIT, Northwestern,

and Rochester for helpful comments. All errors are ours.

The Anderson School at UCLA, Los Angeles, CA 90095-1481, USA. e-mail address:

[email protected]

Department of Economics, UCLA, Los Angeles, CA 90095-1477, USA

Department of Finance, HKUST, Clear Water Bay, Hong Kong

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Abstract

We examine optimal capital allocation and managerial compensation in a firm

with two investment projects (divisions) each run by a manager who can provide

(i) (unverifiable) information about the quality of either or both projects and

(ii) (unverifiable) access to valuable resources that can enhance the cash flows

of either or both projects. In our model, each managers optimal compensation

contract is linear in the cash flows of both projects. The firm optimally provides

more capital, greater performance pay, and a lower salary to a division manager

when she reports (truthfully, in equilibrium) that her project is higher quality.

The firm generally underinvests in capital and managers underinvest resources in

both projects (relative to the first-best level). We derive numerous cross-sectional

predictions about the severity of the underinvestment problem, the sensitivity ofinvestment in one division to the cash flows and investment opportunities of other

divisions, and the relative importance of division-level and firm-level performance-

based pay. We also compare investment policies in multi-division firms to single-

division firms.

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1 Introduction

The long-term health of afi

rm is determined by the quality of its investments. In a multi-division firm, capital is allocated to investment projects based on reports by division

managers who have access to private information about project quality. The eventual

success of any project may also require the input and cooperation of other managers

who control access to valuable resources within the firm.1 The firm, however, may not

be able to either independently verify the managers reports or monitor the actions of

managers to ensure that they are deploying and sharing their resources appropriately.

Thus, to secure its long-term health, the firm must provide incentives to ensure truthful

information flow and the efficient use of its resources.

In this paper, we present a simple model to illustrate these information and incentiveproblems in a multi-division firm. Specifically, we consider a firm with unlimited access

to capital and two investment projects. The optimal amount of capital to allocate to

each project depends on its quality which is unknown to the firms headquarters. For

each project, however, the firm can hire a risk-neutral manager with private information

about her own projects quality and potentially the other projects quality. Once hired,

each manager reports her information (not necessarily truthfully) to headquarters which

then allocates capital according to the reports. The veracity of the reports is assumed

to be non-verifiable and non-contractible. Once capital is allocated, each projects cash

flows can be enhanced by either or both managers by deploying critical resources undertheir control. The use of these resources is assumed to be privately costly, non-verifiable,

and non-contractible.

To fix ideas, consider a firm with two divisions selling computer hardware and ser-

vices. Each division has a manager with private information about the demand for its

product obtained from extensive time in the field. The demand for hardware is related

to the demand for services (and vice-versa) so each managers private information is

relevant for determining the quality of investment projects in both divisions. Moreover,

1The fact that the success of investments in one division also depends on the resources available in

other divisions may very well be the reason the divisions exist within the same firm. However, we do

not explicitly analyze the boundaries of the firm in this paper; rather, we take as given the scope of

the firm. We also assume the firm has access to capital but division managers do not, so the firms

headquarters is indispensable to the production process. Gertner, Scharfstein, and Stein (1994) and

Stein (1997) explicitly model the productive role of headquarters to help understand the costs and

benefits of internal and external capital markets.

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each manager controls resources that are valuable to both projects; for example, the

hardware division manager may have relationships with buyers that can be exploited to

sell services (and vice-versa). The use of this relationship capital with buyers is costlyto the manager and likely to be non-verifiable by the firm.

The firms problem is to design managerial compensation contracts and a capital allo-

cation schedule to encourage truthful reporting and the appropriate provision of critical

resources. In our model, the optimal managerial compensation contract is linear in both

divisions cash flows. Moreover, the firm provides more capital, greater performance-

based pay, and a lower salary to a division manager when she reports (truthfully, in

equilibrium) that her project is higher quality. The lower salary combined with higher

performance-based pay effectively encourages truthtelling by making the manager buy

shares in the firm when she is very optimistic.In general, the firm invests too little capital and managers allocate too few resources

(relative to first-best) to both divisions and this underinvestment problem is more severe

when both divisions have relatively poor investment opportunities, the asymmetric infor-

mation between headquarters and both division managers is greater, division managers

have more firm-specific human capital, and division managers have less performance-

based pay. We also show that investment in one division is positively related to the

quality of investment opportunities in other divisions and this relation is stronger when

managerial resource sharing is more important, division managers have more discretion,

and division managers have more firm-specific human capital. These predictions suggest

refinements to the empirical literature examining the sensitivity of investment in one

division to the cash flows and investment opportunities of other divisions in the firm

(e.g., Lamont, 1997; Shin and Stulz, 1988). The excess sensitivity of investment to the

cash flows of other divisions implies that multi-division firms will invest more (less) in a

division than a single-division firm when other divisions in the firm are expected to per-

form well (poorly). Finally, we derive novel implications for the composition of division

manager compensation contracts. For example, consistent with the empirical evidence

in Bushman, et al. (1995) and Keating (1997), we show that division managers receive

relatively more firm-level performance pay compared to division-level performance pay

when they control resources that are more valuable in other divisions.

There is a large theoretical literature examining optimal capital budgeting mecha-

nisms in single-division firms (e.g., Harris and Raviv, 1996; Holmstrom and Ricart i

Costa, 1986; Zhang, 1997; Berkovitch and Israel, 1998; Bernardo, Cai, and Luo, 2001;

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Garcia, 2001, 2002). This paper extends the analysis in Bernardo, Cai, and Luo (2001)

to multi-division firms with productive and information interactions among divisions.

Harris, Kriebel, and Raviv (1982) and Antle and Eppen (1985) present models of multi-division firms in which the division manager has private information about the pro-

duction technology and a preference for capital. These papers do not consider optimal

managerial compensation mechanisms but rather focus on the role of transfer prices in

allocating capital across divisions. Harris and Raviv (1998) present a model of a single-

division firm with multiple investment projects in which the division manager has private

information and a preference for capital. Headquarters can learn the information through

a costly audit. The optimal mechanism trades off the distortion due to decentralized

information and managerial preference for capital against the costs of (endogenously

determined) probabilistic auditing. As in Harris, Kriebel, and Raviv (1982), they findregions of under- and over-investment, whereas we find only under-investment. These

different predictions follow from their assumption of exogenous compensation contracts

(see Bernardo, Cai, and Luo, 2001). Finally, Rajan, Servaes, and Zingales (2000) pro-

pose a theoretical model of internal capital markets in which division managers exhibit

rent-seeking behavior. In contrast, we consider agency costs due to asymmetric informa-

tion and managerial moral hazard. Their model assumes managerial incentive schemes

are exogenously determined whereas we derive jointly the optimal capital budgeting and

managerial compensation mechanism.

The remainder of the paper is organized as follows. Section 2 presents our model.

Section 3 derives the first-best capital and managerial resource allocations to be used

as a benchmark for the optimal second-best mechanism derived in Section 4. Section 5

discusses the important features of the second-best mechanism and provides directions

for future empirical work on capital budgeting and managerial compensation. Section

6 examines the robustness of our capital underinvestment result to alternative specifi-

cations for managerial preferences. Section 7 concludes and gives direction for future

research.

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2 The model

We consider afi

rm run by a headquarters acting in the interest of thefi

rms risk-neutralshareholders. Headquarters has unlimited access to capital and two investment projects.

The optimal amount of capital to invest in each project depends on its quality, which is

unknown to headquarters. Headquarters, however, can hire a risk-neutral manager for

each project who can add value in two ways: first, they have information about their

own projects quality (which may also be relevant for determining the other projects

quality) and second, they have access to critical resources which can enhance the cash

flows of either or both projects. The deployment of such resources is assumed to be

privately costly to the manager. Examples of critical resources include key personnel,

scarce and valuable assets (e.g., distribution network), or access to a relationship (e.g.,suppliers and buyers). The classic interpretation of a critical resource in the principal-

agent literature is the managers own effort. The key point is that the use of critical

resources imposes costs on the manager who controls them (e.g., restricts its use in an

alternative pet project, depletes valuable political capital) and cannot be contracted

upon.2

Once hired, each manager is asked to report (simultaneously) her private information

to the headquarters, which then chooses the capital allocation to each project based on

both reports. Specifically, we assume the project cash flows, denoted Vi for project

i = 1, 2, are given by:

Vi = nki + (ei + eji)ki + (ti + tj)ki 0.5k2

i + i,

where the subscript j = 2 (j = 1) when i = 1 (i = 2).

In this specification, ki denotes the amount of capital allocated to project i, ti denotes

the private information of manager i, ei denotes the resources employed by manager i

on her own project i, and eij denotes the resources employed by manager i on the other

project j. The quality of project 1 depends on both t1 and t2 where the parameters

0 determine the relative importance of each managers information. For2Our model builds on Laffont and Tirole (1986, 1993) who also examine the fundamental tradeoff

between moral hazard and asymmetric information. Laffont and Tirole consider the problem of regulat-

ing a single monopoly with unobserved efficiency and non-contractible effort. While many of our proof

techniques follow their arguments closely, our model has many different features (e.g., multiple agents,

capital budgeting) and we are concerned with a very different problem; namely, how information and

resource spillovers across divisions affect capital allocations and incentives within a firm.

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example, if = 1 and = 0 only the manager of project 1 has relevant information

about the quality of project 1. The parameters 0 reflect the potential impact of

each managers resources on the project cash flows. The assumptions and capture the idea that each manager has more impact, via information and resources

under her control, on her own project.3 The i are independent noise terms with mean

zero. Finally, n 0 is a constant that ensures it is always worthwhile to invest some

capital in each project in the socially efficient solution.

The cash flow specification, Vi, has many standard and intuitive features. Capital

and managerial resources are complementary, implying that the marginal products of

each are increasing in the levels of the other. This assumption will be important for the

main results of the paper although the specific functional form is not important. Com-

plementarity is a very reasonable assumption if, for example, one interprets managerialresources as effort. Furthermore, the marginal product of capital is increasing in the

quality of the project, defined as (t1 + t2), which is intuitively appealing and implies

that the headquarters will want to allocate more capital to higher quality projects. The

noise terms i are independently distributed and capture underlying uncertainty about

or measurement errors of project cash flows; we show below that since all agents are risk

neutral the additive, mean zero noise terms have no effect on our results.

Headquarters does not know the information, ti, but only knows that ti is drawn

from the interval [0, t] according to a distribution Fi(t) with density function fi(t), where

fi(t) > 0 t. We assume that t1 and t2 are independently distributed and that the ti and

i are independently distributed. As is standard in the mechanism design literature, we

also assume that the inverse of the hazard rate ofFi(), denoted i(t) = (1Fi(t))/fi(t),

is decreasing in t. It is well known that many common distributions such as the uniform

and (truncated) normal distribution have increasing hazard rates.

The deployment of managerial resources imposes costs on the manager. For tractabil-

ity, we assume a specific functional form for the cost function so that the division man-

agers expected utilities are given by:

EUi = Ewi 0.5(e2i + e2ij)

where wi is her compensation, Ewi is her expected compensation, and the parameter

represents the managers private cost of providing (access to) resources under her

3In the special case = = 0 the two projects are completely independent and we can analyze each

firm separately as in Bernardo, Cai, and Luo (2001).

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control.4 In Section 6, we examine the effect of including managerial preferences for

capital. Finally, we assume that each manager has outside employment opportunities

offering the reservation utility U 0.The headquarters problem is to maximize the expected payoff to shareholders, the

residual claimants of the cash flows from the two projects. We assume there are no

conflicts of interest between headquarters and shareholders because these issues are not

central to our thesis. Headquarters can use two instruments, capital and compensation,

to provide incentives for the division managers to tell the truth about project quality

and provide appropriate access to resources under their control. Specifically, headquar-

ters designs an optimal mechanism consisting of (i) a capital allocation policy ki(t1, t2)

depending on each of the division managers reports about project quality, ti, and (ii)

a compensation schedule wi(t1, t2, V1, V2) depending on both reports and both projectoutcomes. Importantly, we assume that the private information, ti, and the managers

resource allocations, ei and eij , are not directly observable or verifiable by the headquar-

ters ex post, therefore, contracts cannot be written on these directly.

The sequence of moves of the game is as follows:

date 0: Headquarters offers each manager a mechanism {wi(t1, t2, V1, V2), ki(t1, t2)}

date 1: The division managers simultaneously report ti.

date 2: Headquarter allocates capital of ki(t1, t2) to division i.

date 3: Each division manager allocates resources to each project, ei and eij .

date 4: The project cash flows are realized and distributed to shareholders less the com-

pensation wi(t1, t2, V1, V2) paid to each division manager.

4An alternative specification for the cost function is 0.5(e1 + e12)2. However, this specification is

problematic because it always gives corner solutions for the managerial resource allocations. In our cost

specification, we assume no interactions between ei and eij . We have also considered other specifications

which allow for interactions. For example, a more general specification is (e21 + e212 + e1e12), where

< 0 implies that e1 and e12 are complementary and > 0 implies that e1 and e12 are substitutes.

When e1 and e12 are complementary, higher e1 reduces the marginal cost of e12 and hence induces

higher e12. Intuitively, one expects to see more convergence between divisions in this case. When

e1 and e12 are substitutes, higher e1 increases the marginal cost of e12 and hence leads to lower e12.

Intuitively, one expects to see more divergence between divisions. However, solving for the closed

form solution in this general case is quite involved technically. In this paper, we focus on the simple

case without cost interactions (i.e., = 0).

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We make the standard assumption in these types of models that headquarters can

commit to the capital allocation scheme offered to the managers at date 0. Absent a

commitment device, it would be optimal for headquarters to allocate a different level ofcapital at date 2 than the amount offered at date 0. If the manager knew this, however,

she would not report truthfully. Headquarters commitment could be the result of (un-

modelled) reputational concerns if it intends to play such a game repeatedly (potentially

with other managers) in the future. Finally, as is standard in all models with asymmet-

ric information and risk-neutral agents, it is critical for what follows that each division

manager observes her private information prior to date 0; otherwise, it would be optimal

for the headquarters to sell the firm to the managers because there is no asymmetric

information at the time of contracting and the risk-neutral managers are equally effi-

cient at bearing the project quality risk as the risk-neutral headquarters. One plausibleexample of our timing assumption is that at the time each division manager is hired

or promoted, she has knowledge of external factors (e.g., market demand, competitors

strategies, industry trends, etc.) relevant to the cash flows of projects the firm may

consider in the future. Another possibility is that the firm recently acquired one of the

divisions and kept its manager to run it in the future. The firm must then choose the

appropriate capital budget for the division and the managers compensation with the

understanding that the manager already has private information about the quality of

the divisions existing and potential investment projects.

3 Benchmark case: First-best outcome

To provide a benchmark, we first determine the socially efficient (first-best) solution of

the model. The first-best maximizes the expected total surplus (expectation over i):

maxk1,k2,e1,e2,e12,e21

E{1}V1 + E{2}V2 0.5(e2

1+ e2

12) 0.5(e2

2+ e2

21).

Before we proceed, we make the following parameter assumptions throughout the

paper:

(A1) > 2 + 2;

(A2) n

1

vuut1 22( 2)

(+ )

0 + t

vuut1 22( 2)

,

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where 0 max(1(0), 2(0)). Assumption (A1) requires that the cost to the manager

of using critical resources () is large relative to the marginal impacts ( and ) of man-

agerial resources on the project cash flows. Technically, this ensures that the marginalcost of using managerial resources is increasing relatively fast so that the objective func-

tion of the optimal mechanism design program is concave. Assumption (A2) requires

that (i) the expected net cash flow from each project is sufficiently high (large n); or (ii)

asymmetric information is not too severe (small and ); or (iii) resource spillovers are

relatively strong (large relative to 2). Technically, Assumption (A2) ensures that

the solution to the mechanism design program yields interior capital allocations (ki > 0)

and resource allocations (ei > 0 and eij > 0). For example, if n is large enough the firm

will always want to allocate at least some capital and some profit-sharing to motivate

managers to provide resources to each project. This assumption allows us to focus onthe most interesting parameter region without having to consider cases where one or

more choice variables is at a boundary. None of the qualitative results of the model are

affected by this assumption.

Proposition 1. Thefirst-best capital allocations and efforts are given by:

kFBi = (12

2

)1(n + ti + tj)

eFBi =

kFBi

eFBji =kFBi

The proof is in the Appendix. It is intuitively clear and straightforward to show that,

in the first-best outcome, the capital allocations and managerial resource allocations in-

crease in and (importance of managerial resources), increase in and (importance

of project quality), increase in t1 and t2 (information about project quality), and de-

crease in (managers private cost of resource provision). If headquarters could observet1 and t2, and the managerial resource allocations, ei and eij , then it should write a

complete contract with each division manager specifying the capital allocations and re-

source choices described in Proposition 1. The salary should be set to levels satisfying

the division managers participation constraint.

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4 Second-best outcome

We now solve for the headquarters optimal mechanism, under the assumption thatit cannot contract on the managers private information, t1 and t2, or the managers

resource allocations, ei and eij . By the Revelation Principle we can, without loss of

generality, restrict our attention to direct revelation mechanisms in which both division

managers report their information truthfully. Thus, the headquarters mechanism design

problem can be stated as:

maxwi,ki,ei,eij

Zt0

Zt0

Z1

Z2

[V1 + V2 w1 w2]dG2dG1dF1dF2

such that

(i) {ei, eij} arg max E{1,2}wi(t1, t2, V1, V2) 0.5(e2

i + e2

ij), (IC1)

(ii) ti arg max E{tj ,1,2}Ui(ti, ti), (IC2)

(iii) {t1, t2}, E{tj ,1,2}Ui(ti, ti) U, (IR)

(iv) {t1, t2}, ki, ei, eij 0. (NN)

where Gi is the distribution ofi; E{1,2} denotes expectations over the random variables

{1, 2}; and Ui(ti, ti) is the utility of manager i who reports ti, has true type ti, and

assumes that the other manager is reporting her true type. Specifically, Ui(ti, ti)

wi(ti, tj, V1, V2) 0.5(e2

i + e2

ij), where Vi = Vi(t1, t2, ki(ti, tj), ei, eij, ej, eji , i) and all the

resource allocations, ei and eij , are ex post optimal for the division managers.

The first incentive compatibility constraint (IC1) requires that the division managers

resource allocations are ex post optimal. The second incentive compatibility constraint

(IC2) requires that it is optimal for each division manager to report truthfully given

that the other division manager will also report truthfully. The constraint (IR) is the

standard interim individual rationality constraint, requiring that for each division man-

ager type ti, her expected equilibrium payoff should be at least as large as her outside

reservation utility, U. The last constraint (NN) requires that capital allocations and

resource allocations are non-negative.

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Proposition 2. The headquarters maximum expected payoff is

E = 0.5E{t1,t2}"(n + t1 + t2 1 2)2

1 2/ 2/ +

(n + t2 + t1 2 1)2

1 2/ 2/

+(21

+ 22)(2

2+

2

2)

# 2U.

The optimal mechanism can be implemented in dominant strategies by the following

capital allocation policy and linear compensation scheme:

ki = (12

2

)1(n + ti + tj i j),

wi = ai + biVi + bijVj

where

bi = 1i2ki

, bij = 1i2kj

,

ai = U +Zti0

(biki + bijkj)ds + 0.5(e2

i + e2

ij)

bi[nki + (ei + eji)ki + (ti + tj)ki 0.5k2

i ]

bij[nkj + (ej + eij)kj + (tj + ti)kj 0.5k2

j ].

The optimal resource allocations implemented are:

ei =ki

i

, eij =kj

i

.

The proof is in the Appendix.5

The optimal mechanism can be implemented by a linear managerial compensation

contract consisting of salaries, ai, shares of own-division cash flows, bi, and shares of

other-division cash flows, bij ; and capital allocation schedules, ki. Moreover, since the

optimal mechanism is implementable in dominant strategies, it is not in the interest

of the division managers to collude even if they know each others private information

5Our proof follows closely the arguments of Laffont and Tirole (1986, 1993), although our model is

more general (includes multiple agents and capital budgeting). Laffont and Tirole (1993, pp. 68-73

and pp. 171-172) provides sufficient conditions for the optimality of linear contracts in a model of the

optimal regulation of a monopoly. See also McAfee and MacMillan (1986) and Holmstrom and Milgrom

(1991) for the description of other settings in which the linear contract is optimal.

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(as long as they cannot write binding side contracts). Finally, it is interesting to note

that the optimal contract is unaffected by the noise terms, i. The reason is that the

managerial contract trades off the benefit of providing incentives to deploy resourcesefficiently against the costs of eliciting truthful reporting. In our model, the additive

noise term does not affect either (i) the fundamental tradeoffbetween moral hazard and

asymmetric information (e.g., if the noise term 1 0 and e21 and t2 are fixed then

observing V1 will allow the headquarters to infer e1 +t1 but not e1 or t1 separately) or

(ii) the cost of providing incentives when all agents are risk-neutral. In contrast, noise

terms do affect the fundamental tradeoff between providing incentives and risk-sharing

in models with moral hazard and risk-averse agents and thus affect the optimal contract

in these settings (e.g., the informativeness principle). We include the noise term because

it shows that our results hold more generally (deterministic cash flow is a special case)and it is natural to assume that cash flows are subject to measurement errors and other

random shocks.6

The following corollary illustrates some important features of this mechanism.

Corollary 1. Monotonicity and comparative statics

(i) The mechanism {ai(t1, t2), bi(t1, t2), bij(t1, t2), ki(t1, t2)} and managerial resource

allocations, ei(t1, t2) and eij(t1, t2), are continuous and monotonic over the whole

domain [0, t]X[0, t]. The profit-sharing rules, bi(t1, t2) andbij(t1, t2), the capital al-

locations, ki(t1, t2), and the managerial resource allocations, ei(t1, t2) andeij(t1, t2)

are non-decreasing in each argument. The salary ai(t1, t2) is non-increasing in ti.

(ii) The capital allocations, ki, increase with , , t1, t2, and n; decrease with ; and

increase with () when ti (tj) is high and decrease when ti (tj) is low.

(iii) The performance-paybi (bij) increases with, , t1, t2, and n, and decreases with

and ; bi increases (decreases) with when tj is high (low); and bij increases

(decreases) with when tj is high (low).

(iv) The managerial resources ei (eij) increase with , , t1, t2, and n, and decreasewith ; ei increases in () when ti (tj) is high and decreases in () when ti

(tj) is low; and eij increases in () when tj (ti) is high and decreases in ()

when tj (ti) is low.

6A more technical proof of the robustness of the linear contract to additive noise is given in Laffont

and Tirole (1993, pp. 72-73) and the references therein.

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The proof is in the Appendix. The optimal mechanism allocates more capital and

greater profit-sharing to the manager when she reports a higher project quality. More-

over, more capital and greater profit-sharing is offered to manager 1 when manager 2reports a higher project quality (and vice-versa). To induce truthtelling, the salary

component of the compensation scheme is lower for higher reported project qualities.

The comparative statics for the optimal mechanism can be understood by considering

the benefits and costs of providing capital and profit-sharing. On one hand, the marginal

benefit of providing capital to project 1 increases in the project quality (t1 + t2) and

in the importance of managerial resources (e1 + e21). Moreover, because managerial

resources allocated to project 1 are given by e1 =b1k1

and e21 =

b21k1

, the marginal

benefit of providing profit-sharing increases in , , and k1, and decreases in . On the

other hand, the marginal cost of providing capital and managerial incentives dependson the cost of maintaining incentive compatibility. Specifically, we show in the proof

of Proposition 2 in the Appendix (see equation (8)) that incentive compatibility for

manager 1 (a similar condition holds for manager 2) requires:

EU1(t1) = U +Zt10

Zt0

(b1k1 + b12k2)dF2ds1. (1)

This states that to induce truthtelling, manager 1 of type t1 must receive expected

utility of EU1(t1) as in equation (1). The term

Rt10

Rt0

(b1k1 + b12k2)dF2ds1 represents

the type-t1 managers information rents. These information rents are increasing in t1.Truthtelling can be achieved with any contract yielding the manager EU1(t1); however,

the marginal cost of providing managerial incentives, b1 and b12, and capital, k1 and

k2, spills over into the information rents that must be paid to all managers with higher

quality projects. This increases the marginal cost of offering profit-sharing and capital

allocations (especially for low-t managers). Moreover, these costs are increasing when

asymmetric information is more important (high and ).

Thus, an increase in quality, t1 and t2, increases (decreases) the marginal benefit

(cost) of capital and profit-sharing; therefore, capital, managerial resources, and profit-

sharing will increase. An increase in the impact of managerial resources, and ,increases the marginal benefit of capital and profit-sharing; therefore, capital, managerial

resources, and profit-sharing will increase. On the other hand, an increase in the resource

cost parameter, , increases the marginal cost of deploying managerial resources (and

allocating capital since managerial resources and capital are complementary); therefore,

capital, managerial resources, and profit-sharing decrease. Finally, an increase in the

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importance of private information, and , increases the marginal benefit of capital;

however, it also raises the marginal cost of providing capital and managerial incentives,

especially for low-quality projects. Thus, for high-project qualities the former effectdominates and capital allocations increase whereas for low-project qualities the latter

effect dominates and capital allocations decrease.

The information rents specification in equation (1) highlights the importance of both

asymmetric information and moral hazard in our model. If there is no asymmetric

information ( = = 0) headquarters can choose full profit-sharing and the first-best

level of capital without increasing managerial information rents. Moreover, if there

is no moral hazard ( = = 0) headquarters can implement first-best by choosing

profit-shares equal to zero and setting capital to their first-best levels without increasing

managerial information rents.

5 Interpretation and Empirical Implications

For what follows it will be useful to interpret the key paramaters in our model: , , ,

, and . The parameter represents both the importance of unverifiable managerial

resources for the managers own project cash flows and the degree of complementarity

between these resources and capital, thus we expect to be greater, for example, when

managers have firm-specific human capital. The parameter represents the importance

of managerial resources for the cash flows of other projects in the firm, thus we expect

to be smaller in more diversified firms. The parameters () represent the importance of

the division managers information for her own (other) projects cash flows. We expect

to be greater when asymmetric information between headquarters and the managers

is more severe (e.g., R&D on a new drug or technology), and to be greater when

there are common factors affecting project cash flows (e.g., market demand in adjacent

sales regions, industry trends). Finally, the parameter represents the cost to the

manager of using (unverifiable) resources under her control; however, an alternative and

more empirically useful interpretation of 1/ (more generally, the inverse of the secondderivative of the cost function 1/C00(e)) is the responsiveness of unverifiable actions to

an increase in incentives (Milgrom and Roberts, 1992). In this interpretation, is higher

when the manager has less discretion over job tasks.

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5.1 Investment in Capital and Managerial Resources

The following implication demonstrates that in the second-best mechanism there is un-

derinvestment in capital and underutilization of managerial resources.

Implication 1:

(i) The capital allocation is lower than the first-best solution. For each division, the

underinvestment in capital is most severe for low-quality projects, becomes less

severe when either managers report increases, and vanishes for the highest possible

quality projects. For a given project quality, the underinvestment decreases in ,

and increases in , , and .

(ii) Managerial resources are lower than the first-best solution. For each division, theunderinvestment in resources is most severe for low-quality projects, becomes less

severe when either managers report increases, and vanishes for the highest possible

quality projects. For a given project quality, the underinvestment of resources in

the managers own project decreases in , increases in , and , and is ambiguous

in ; and the underinvestment of resources in the other project decreases in ,

increases in , and , and is ambiguous in .

The proof is in the Appendix. As we demonstrated above, incentive compatibil-

ity (in the second-best mechanism) requires that when the firm increases its capitalallocation to any manager it must also increase the information rents to all higher-

type managers. This makes the marginal cost of providing capital high, especially for

low-quality projects. Consequently, there is underinvestment of capital in the optimal

mechanism (relative to the first-best) and the underinvestment problem is more severe

for lower quality projects. By allocating less capital and less profit-sharing to low-quality

projects, headquarters reduces the incentive costs for high-quality projects where it mat-

ters most. From equation (1) we see that the marginal cost of providing capital is more

severe when and are larger, thus the underinvestment problem is more severe when

there is more asymmetric information. Moreover, the marginal cost of providing capitalis higher when profit-sharing is higher and the latter increases in and and decreases

. Thus, the underinvestment problem is more severe when managerial resources are

more important and when the manager has more discretion. Finally, managerial re-

source allocations are increasing in the managers profit share and the capital allocated

to each project. However, since the second-best mechanism provides too little capital

14


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and less-than-full profit-sharing, there is also underinvestment in managerial resources

relative to the first-best.

Our underinvestment result contrasts with Harris and Raviv (1996, 1998) and Harris,Kreibel, and Raviv (1982), who found that overinvestment occurs for the lowest quality

projects while underinvestment occurs for the highest quality projects.7 Furthermore,

our result is consistent with the evidence that firms adopt higher hurdle rates of re-

turn than predicted by standard finance theory (Poterba and Summers, 1992). Several

other theories are consistent with this prediction including, for example, real options

models in which there is value to waiting to invest in a project. However, our model

makes further predictions which distinguish our theory from the others. For example,

we predict that the underinvestment problem in a given division is more severe when

(i) asymmetric information between headquarters and the division manager is greater(larger and ), (ii) the division is more human-capital intensive (larger ), and (iii)

the other divisions in the firm are performing poorly. For example, we predict that the

difference between observed hurdle rates and predicted hurdle rates will be large for

companies with high R&D expenses because project managers/scientists in these firms

typically have considerable private information and firm-specific human capital.

Moreover, the underinvestment problem is associated with the division manager re-

ceiving less performance-based pay and managers of other divisions in the firm receiving

less performance-based pay. Consistent with this prediction, Palia (2000) finds empirical

evidence that underinvestment in relatively strong divisions of diversified firms is less se-

vere when the division managers have a higher percentage of the firms equity in options

and shares. Interestingly, in our model division managers receive greater performance-

based pay because they manage higher quality projects, not necessarily that greater

performance-based pay causesfirm value to increase. Thus, we argue that one must be

careful interpreting results such as those found in Palia (2000) since the theory does not

suggest an obvious causal link between performance-pay and eventual performance.

7We demonstrate in an earlier paper (Bernardo, Cai, and Luo, 2001) that this difference results from

their assumption of exogenous compensation contracts.

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Implication 2:

(i) The capital allocated to each project is more sensitive to the other managers

information in the second-best solution than in the first-best solution.

(ii) The difference in sensitivity to the other managers information is stronger for

higher , , and ; and lower .

The proof is in the Appendix. The intuition for result (i) is that positive information

about t2 has two effects on the capital allocated to project 1: first, high t2 also implies

that project 1 has better prospects so more capital is allocated to it, and second, since

manager 2 is a higher-type the firm optimally provides higher-powered incentives for

her to provide resources to both projects. The extra resources deployed by manager 2 inproject 1 increases the marginal product of capital in project 1. The first effect is present

in both the first-best and second-best mechanisms whereas the second effect provides a

motivation for increasing the capital allocation to project 1 not present in the first-best

solution. Consequently, the optimal amount of capital to allocate to project 1 is more

sensitive to the quality (cash flows) of project 2 in the second-best mechanism.

The sensitivity of investment in one division to the cash flows and investment op-

portunities in other divisions in the firm has been studied extensively (see, e.g, Lamont,

1997; Shin and Stulz, 1998; Rajan, Servaes, and Zingales, 2000). For example, Shin

and Stulz argue that we should expect to see investment in one division falling as theinvestment opportunities (measured by industry-Q) in other divisions improve if inter-

nal capital markets are redirecting capital to its best use. They find, however, that

investment-to-asset ratios in divisions of conglomerate firms are relatively insensitive

(slightly negative) to the investment opportunities of other (ostensibly unrelated) di-

visions and interpret this to mean that internal capital markets are inefficient. A key

element of their argument is that the firm is capital constrained. By contrast, in our

model the firm does not face a capital constraint and, as a result, we argue that we should

expect investment in one division to be insensitive to the investment opportunities of

other divisions if these other divisions investment opportunities are independent (as

Shin and Stulz observed). Moreover, we predict the coefficient in a Shin-Stulz type re-

gression of the investment-to-asset ratio in one division on the investment opportunities

in another division (typically measured by industry Tobins q) should be greater when

divisions are more related (larger and ), managers have more discretion (smaller ),

and managers have more firm-specific human capital (larger ).

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5.2 Managerial Compensation

For what follows, we re-write the managers wage contract:

wi = ai + biVi + bijVj ai + bij(Vi + Vj) + (bi bij)Vi.

Thus, we can interpret the coefficient bij as the division managers firm-level performance-

based pay and (bi bij) as the division managers division-level performance-based pay

when (bi bij) > 0. The following implication follows immediately from Corollary 1:

Implication 3: Division managers receive greater firm-level performance pay when

(i) project quality is higher (higher ti), (ii) managerial resources are more valuable

in their own division (larger ), (iii) managerial resources are more valuable in other

divisions (larger ), and (iv) division managers have more discretion (smaller ).

The following implication considers the use of division-level performance pay both

in absolute terms and in relation to firm-level performance pay (i.e., bibijbij

).

Implication 4: Holding division size constant (k1 = k2), division managers receive

less division-level performance pay in both absolute terms and relative to firm-level

performance pay when (i) managerial resources are more valuable in other divisions

(larger ) and (ii) the division managers own private information is more important for

her projects cash flow (larger ).

The proof is in the Appendix. The intuition for part (i) is that when managerial

resources are valuable in many divisions the firm wants to provide incentives for the

manager to share these resources by de-emphasizing division-level performance pay. The

intuition for part (ii) is that when is large, information rents are high to induce

truthful reports on the division managers own division thus the firm again de-emphasizes

division-level performance pay. Our implication is consistent with the empirical work of

Bushman, et al. (1995) who find that the use offirm-level performance pay is increasing

when divisions (groups) are more related whereas unrelated division manager pay is

more closely related to division performance. Keating (1997) also finds that the use offirm accounting metrics in division manager compensation is positively related to the

impact the manager being evaluated has on other divisions in the firm.

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Implication 5: Firm-level performance-based pay is positively correlated across

divisions.

The proof is in the Appendix. The intuition for this result is as follows. If, for

example, manager 1 reports a high t1 then both projects have higher quality and the firm

optimally allocates more capital to both divisions. Because managerial resources and

capital are complementary this increases the marginal benefit to the firm of providing

incentives for resource sharing (increased bij) by offering firm-level performance pay.

Consistent with this result, Palia (2000) finds that firm-level performance pay (measured

by the share offirm equity) is positively correlated across divisions.

5.3 Comparison of multi-divisionfi

rm to single-divisionfi

rmWe now compare capital allocations in single-division and multi-division firms. To do

so, we assume that the single division firm enjoys some market-average resource spillover

e21 and can only contract on its own net cash flows (b12 = 0). Moreover, it does not

observe a precise report t2 but observes some market signal of t2, t2. As an example

of the resource spillover, e21, consider a firm consisting of an auto assembly division

and an auto engine division that splits into two single-division firms. Even though the

two separate firms may not be able to co-ordinate and utilize the resource spillovers

as well as the headquarters of the two-division firm can, they probably maintain close

business relationships through market transactions and some of the spillovers should still

be realized.

Under these assumptions, it can be shown that the optimal capital allocation in the

second-best mechanism for the single-division firm is given by:

k11

=

2(n + (t1 1) + t2 + e21).

We are interested in comparing investment policies in a multi-division firm to a single-

division firm holding project quality constant. We implement this strategy by holding

t1 fixed and assuming the multi-division firm receives a report t2 = t2. Comparing

investment in the single-division firm, k11

, to the investment in division 1 of the multi-

division firm, k1 (in Proposition 2), we get the following implication.

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Implication 6: The multi-division firm invests more (less) in a division than a

single-division firm when other divisions in the multi-division firm are performing well

(poorly). On average, the multi-division firm invests more than the single-division firmwhen managerial resources are more valuable in other divisions (larger ).

The proof is in the Appendix. The first statement follows closely from our Implication

2. The intuition for the second statement is that each division in the multi-division firm

derives benefits from using resources in the other division (and these benefits increase

in ) but also pays greater information rents in order to get the manager of the other

division to report truthfully. For large , the resource benefits exceed the information

rent costs.

6 Alternative Managerial Preferences

We now check the robustness of our results to alternative specifications for managerial

preferences. In our model, we chose to consider the importance of asymmetric infor-

mation and managerial moral hazard. Another well-accepted and reasonable conflict

that might emerge between shareholders and management is managerial preference for

capital. This may reflect enhanced reputation from controlling bigger projects, a prefer-

ence for empire-building, or greater perquisite consumption that comes from running

a larger business. One reasonable specification is to assume that the manager derivesutility from monetary rewards and from controlling large (high k), high quality (high t)

projects so that managerial preferences are given by:

EUi = Ewi 0.5(e2

i + e2

ij) + (ti + tj)ki,

where 0 measures the degree of managerial preferences for capital. Our original

model corresponds to the special case of = 0.

With these modified managerial preferences and everything else being identical to

our earlier model, it is straightforward to show that the first-best capital allocation is:

kFBi = (12

2

)1[n + (1 + )ti + (1 + )tj].

The first-best capital allocation increases in the managerial preference for capital, ,

because the headquarters can internalize the managers preference perfectly by reducing

her salary (negative compensation may be needed).

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With asymmetric information and moral hazard, it can be shown that maintaining

incentive compatibility in the second-best mechanism requires:

EUi(ti) = U +Z

ti

0

Zt

0

((bi + )ki + bijkj)dFjdsi.

Comparing this to equation (1) we see that the marginal cost of providing capital

increases with . Applying the same method of solving for the optimal mechanism, the

second-best capital allocation is given by:

ki = (12

2

)1[n + (1 + )(ti i) + (1 + )tj j].

Thus, underinvestment in the second-best solution is

kFBi ki = (12

2

)1[(1 + )i + j ].

Clearly, kFBi ki is increasing in . Moreover, since

kitj

kFBitj

= (12

2

)1

djdtj

> 0,

from Implication 2 we can see that the (over-)sensitivity of investment in one division to

the other managers information (relative to thefi

rst best) is unchanged with preferencesfor capital. Summarizing we have the following result:

Proposition 3. Managerial preference for capital exacerbates the underinvestment prob-

lem but has no effect on the cross-division (over-)sensitivity of investment to information

(relative to the first- best solution).

The discussion above indicates that managerial preference for capital has only sec-

ondary effects on capital budgeting, and its effects operate through its impact on the

optimal incentive scheme to control for asymmetric information and moral hazard.8 This

point can be made clearer if we modify the managers preferences for capital and ignore

the project quality:

EUi = Ewi 0.5(e2

i + e2

ij) + ki.

8A similar conclusion holds when managers have preferences for managing a relatively large and

high-quality division.

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In this case, managers have no control over the control benefits (ki), thus the costs of

maintaining incentive compatibility are identical to our main model in which managers

have no preferences for capital. With this specification, headquarters can internalizemanagerial preferences for capital perfectly even in the second-best solution and the

underinvestment problem will not be exacerbated.

7 Conclusions

Firms make capital budgeting decisions based on the reports of project (division) man-

agers. In many cases, these reports can neither be independently verified before signifi-

cant capital investments are made nor contracted upon ex post. Moreover, the eventual

success of these projects may also depend on the (unverifiable) energy and resources de-

voted to it by managers throughout the firm. In such cases, the firm must put in place

explicit incentives for managers to provide truthful reports and to utilize and share

resources efficiently.

In this paper, we examined the extent to which capital budgeting and managerial

compensation mechanisms can mitigate these information and incentive problems in a

multi-division firm. In the optimal mechanism, firms underinvest in capital and man-

agers underinvest resources in all divisions. Moreover, this underinvestment problem in

one division is more severe if the division has poor prospects, the other divisions in the

firm have poor prospects, asymmetric information between headquarters and all divi-

sion managers is more severe, and managers have a preference for capital (i.e., managing

larger divisions).

We believe our model is best suited to cases where division managers request capital

infrequently (e.g., R&D on a new drug). In such cases, compensation schemes such as

shares in the firm (with restrictions on selling) or stock options (with a long vesting

period) are likely to provide powerful incentives for managers to report truthfully their

private information and allocate efficiently the resources under their control. However,

our model is not well-suited to cases in which division managers request capital fre-quently. In such cases, reputation effects provide powerful incentives because the firm

can always deny the manager future capital requests if her earlier analysis is offthe mark.

In future research, we would like to determine the extent to which such reputation effects

substitute for explicit managerial incentives in the capital budgeting process.

There are many other potential directions for future research. In our model, the

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multi-division firm is always better off than a portfolio of single-division firms because

it can contract on more variables and thus mitigate incentive problems at a lower cost.

However, in reality there may be countervailing costs to firm scope (e.g., managerialrent-seeking behavior) which are important for determining the optimal boundary of

the firm. It would also be interesting to examine the optimal degree of centralization in

a resource-constrained firm. For example, suppose each division manager has preferences

for capital and multiple investment projects. If the firm allocates, say, 1/N resources to

each of the N divisions then each division manager has the incentive to choose the best

projects in her division. However, if the capital budgeting process is centralized, the firm

might incur potentially large contracting costs to induce truthful reports from division

managers about the appropriate allocation across divisions. Thus, the optimal degree

of centralization trades-off intra-division efficiency against inter-divisionefficiency.

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Appendix

Proof of Proposition 1. In the first-best, headquarters maximizes total expected surplus:

nk1 + (e1 + e21)k1 + (t1 + t2)k1 0.5k2

1

+nk2 + (e2 + e12)k2 + (t2 + t1)k2 0.5k2

2

0.5(e21 + e2

12) 0.5(e2

2 + e2

21).

The first-order (necessary) conditions are (taking derivatives with respect to ki, ei, and eij , respectively):

0 = n + ei + eji + ti + tj ki,

0 = ki ei,

0 = kj eij ,

which yields the result. The second-order condition for the total surplus-maximization problem requires

that the Hessian matrix of second derivatives of the objective function is negative semi-definite which

is easy to verify under Assumption (A1). Q.E.D.

Proof of Proposition 2. Our proof strategy for finding the optimal Bayesian-Nash mechanism follows

Laffont and Tirole (1986) and McAfee and McMillan (1987). The proof consists of two steps. In Step 1,

we relax the IC constraints in such a way that it is possible to derive the optimal capital allocation and

compensation mechanism and hence the headquarters expected payoff. Since this relaxed program hasfewer constraints than the original program, the headquarters expected payoff in this relaxed program

provides an upper bound of the value attainable in the original program. In Step 2, we consider a

narrower class of mechanisms than in the original program. Specifically, we focus on mechanisms with

linear compensation schemes. Clearly the headquarters expected payoff in this class provides a lower

bound of the value attainable in the original program. We then demonstrate that this lower bound is

identical to the upper bound derived in Step 1 and thus the headquarters optimal mechanism can be

implemented with the linear compensation scheme.

Step 1: Let ei (t1, t2) and e

ij(t1, t2) represent the headquarters desired resource allocations for truthful

reports {t1, t2}. Suppose that headquarters could observe the actual values ofei+ti and eij +ti for

both division managers and suppose manager j reports tj . If division manager i reports ti, headquarters

wishes to see ei (t1, t2)+ti and e

ij(t1, t2)+ti; otherwise, she knows that one of the division managers

has lied about her information or has not followed the recommended resource allocation policies, and

hence will punish (with arbitrary severity) the managers for their deviations. Thus, conditional on

manager j reporting tj , if division manager i reports ti, she will have to choose ei and eij to be

consistent with her report; that is, ei + ti = ei (t1, t2) + ti and eij +ti = e

ij(t1, t2) +ti, which

yields:

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ei = e

i (t1, t2) + (ti ti)/,

eij = e

ij(t1, t2) + (ti ti)/. (2)

Note that by Equation 2, if the division managers report truthfully {t1 = t1, t2 = t2}, they must

follow headquarters recommended resource allocation policies. Thus the (IC1) constraint is completely

relaxed.

Denote the headquarters original problem as (P1) and now consider the following problem (P2):

maxwi,ki,ei,eij

Zt0

Zt0

Z1

Z2

V1 + V2 w1 w2

dG2dG1dF1dF2

such that

(i) ti arg maxE{tj,1,2}Ui(ti, ti), (IC2)

(ii) {t1, t2}, E{tj,1,2}Ui(ti, ti) U (IR)

(iii) {t1, t2}, ki, ei, eij 0 (NN)

where Ui(ti, ti) = wi(ti, tj , V1, V2)0.5(e2i +e

2ij) is the utility to manager i who reports ti, has true type

ti, and assumes that the other manager is reporting her true type; Vi = Vi(t1, t2, ki(ti, tj), ei, eij, ej , eji, i);

and ei = ei, eij = eij , ej = ej(ti, tj), and eji = eji(ti, tj).

Program (P2) replaces the (IC1) constraint of Program (P1) with Equation 2 and incorporates it

in the (IC2) constraint, so (P2) is a relaxed program of (P1). Ignoring the (NN) constraint for the

moment, we now solve (P2).

Given Equation 2, for any true types (t1, t2) and any reports {t1, t2}, the values of cash flows V1

and V2 can be expressed as:

V1(t1, t2, 1) = nk1(t1, t2) + (e

1(t1, t2) + e

21(t1, t2))k1(t1, t2)

+(t1 + t2)k1(t1, t2) 0.5k1(t1, t2)2 + 1

V2(t1, t2, 2) = nk2(t1, t2) + (e

2(t1, t2) + e

12(t1, t2))k2(t1, t2)

+(t2 + t1)k2(t1, t2) 0.5k2(t1, t2)2 + 2.

Thus the cash flows are independent of the true types (t1, t2), which is critical for what follows. Note

also that the compensations may be written as wi(t1, t2, V1(t1, t2, 1), V2(t1, t2, 2)) so we can write

E{1,2}[wi(t1, t2)] = E{1,2}[wi(t1, t2, V1(t1, t2, 1), V2(t1, t2, 2))].

The (IC2) constraint in (P2) now states that conditional on tj = tj ,

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ti arg maxEUi(ti, ti)

= Etj [E{1,2}[w1(ti, tj)]] 0.5Etj

hei (ti, tj) +

(ti ti)

i2 0.5Etj

heij(ti, tj) +

(ti ti)

i2.

By the Envelope Theorem, the (IC2) condition implies

dEUi(ti, ti)

dti=

EUi(ti, ti)

ti|ti=ti +

EUi(ti, ti)

ti|ti=ti

=EUi(ti, ti)

ti|ti=ti =

Etj [e

i (ti, tj)]

+

Etj [e

ij(ti, tj)]

. (3)

Integration yields

EUi(ti) = Ui(0) +Zt0

Zti0

(e

i (s, tj)

+ e

ij(s, tj)

)dsdFj .

By the (IR) constraint, it must be that Ui(0) = U. Taking the expectation with respect to ti yields

EUi = U +

Zt0

Zt0

Zti0

(ei (s, tj)

+

eij(s, tj)

)dsdFjdFi

= U +

Zt0

Zt0

(ei (ti, tj)

+

eij(ti, tj)

)idFjdFi

where i = (1 Fi)/fi.

Substituting the expressions for wages, Ewi = EUi + 0.5E[e2

i + e2ij ], into the expected payoff

yields

E =

Zt0

Zt0

nnk1 + (e

1+ e21)k1 + (t1 + t2)k1 0.5k

2

1 U1 0.5(e2

1+ e2

12)

+nk2 + (e

2 + e

12)k2 + (t2 + t1)k2 0.5k2

2 U2 0.5(e2

2 + e2

21)o

dF1dF2

=

Zt0

Zt0

nnk1 + (e

1 + e

21)k1 + (t1 + t2)k1 0.5k2

1 + nk2 + (e

2 + e

12)k2

+(t2 + t1)k2 0.5k2

2 0.5(e2

1+ e2

12) 0.5(e22 + e

2

21)

(e1

+e12

)1 (e2

+e21

)2odF1dF2 2U . (4)Point-wise differentiation of the integrand gives the following first order conditions

0 = n + ei + e

ji + ti + tj ki,

0 = ki e

i

i,

0 = ki e

ji

j . (5)

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Thus,

ki = (1 2

2

)1(n + ti + tj i j),

ei = 1(ki

i),

eij = 1(kj

i). (6)

The s.o.c. holds for > 2 + 2. The maximized integrand in equation (4) is given by

M11,11 = 0.5

(n + t1 + t2 1 2)2

1 2/ 2/+

(n + t2 + t1 2 1)2

1 2/ 2/

+(21 + 22)( 2

2+

2

2)

.

It is easy to check that {ki, ei , e

ij} are all increasing in (ti, tj). By standard arguments (Mirrlees,

1971), the (IC2) constraint is satisfied. Plugging Equation 6 into E, the headquarters expected payoff

can now be expressed as

EP2 = 0.5E{t1,t2}

(n + t1 + t2 1 2)2

1 2/ 2/

+(n + t2 + t1 2 1)

2

1 2/ 2/+ (21 +

2

2)(2

2+

2

2)

2U . (7)

Under Assumption (A2), it can be checked that the solution {ki, ei , e

ij} given by Equation 6 is

strictly positive for all (t1, t2). This means the (NN) constraint is satisfied and non-binding.

To prove that Equation 6 gives an optimal mechanism for Program (P2), we need to show that it

dominates mechanisms under which the (NN) constraint is at least partly binding. This is necessary

because a binding (NN) constraint may relax the (IC) constraints, as the set of possible deviations for

the managers is reduced. We can immediately rule out the boundary case ki = 0 since capital is critical

to project cash flows.

Consider an arbitrary mechanism such that e1 = 0 and all other es are strictly positive. For this

mechanism, let us completely relax the (IC1) and (IC2) constraints for manager 1. That is, he must

report truthfully t1 and follow the recommended e12. Clearly no information rents are paid to manager

1. Manager 2s incentive constraints can be handled as before, which leads to maximizing

E =

Zt0

Zt0

nnk1 + e21k1 + (t1 + t2)k1 0.5k

2

1+ nk2 + (e2 + e12)k2

+(t2 + t1)k2 0.5k2

2 0.5e2

12 0.5(e2

2+ e2

21) (

e2

+e21

)2

odF1dF2.

It can be easily checked that the maximized integrand in this case is

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M01,11 = 0.5 (n + t1 + t2 2)

2

1 2

/

+(n + t2 + t1 2)

2

1 2

/ 2

/

+ 22(2

2

+2

2

) .Similarly we can derive the maximized integrands in other cases involving binding (NN) constraints:

M11,10 = 0.5

(n + t1 + t2 1)

2

1 2/+

(n + t2 + t1 1)2

1 2/ 2/+ 2

1(2

2+

2

2)

,

M11,01 = 0.5

(n + t1 + t2 1)2

1 2/ 2/+

(n + t2 + t1 1)2

1 2/+ 21(

2

2+

2

2)

,

M10,11 = 0.5

(n + t1 + t2 2)

2

1 2/ 2/+

(n + t2 + t1 2)2

1 2/+ 2

2(2

2+

2

2)

,

M10

,10

= 0.5(n + t1 + t2)

2

1 2/ +

(n + t2 + t1)2

1 2/

,

M10,01 = 0.5

(n + t1 + t2)

2

1 2/ 2/+ (n + t2 + t1)

2

,

M01,10 = 0.5

(n + t1 + t2)

2 +(n + t2 + t1)2

1 2/ 2/

,

M01,01 = 0.5

(n + t1 + t2)

2

1 2/+

(n + t2 + t1)2

1 2/

,

where a 0 in the subscript indicates a binding (NN) constraint, for example {11, 10} means that only

e21 = 0 and {10, 01} means e12 = 0 and e2 = 0.

Under Assumption (A2), it can be checked that M11,11 is greater than all other Ms listed above for

all (t1, t2). Since M11,11 is globally optimal, it is also greater than any combination of other Ms mixed

over different regions of (t1, t2). This implies that the mechanism given by Equation 6 gives greater

expected payoff to the headquarters than any other mechanism that involves binding (NN) constraints.

Therefore, Equation 6 gives an optimal mechanism for Program (P2).

Since (P2) is a relaxed program of (P1), we must have

EP2 EP1

where EP2 is given by Equation 7 and EP1 is the headquarters maximum expected payoff in the

original program (P1).

Step 2: Now we go back to the original program (P1), but restrict our attention to a class of mechanisms

with linear compensation rules:

wi = ai + biVi + bijVj

where {ai, bi, bij} are functions of the reported types {t1, t2}. Substituting E{1,2}wi into manager is

utility function gives:

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Ui = ai + bihnki + (ei + eji)ki + (ti + tj)ki 0.5k2i i+bij

hnkj + (ej + eij)kj + (tj + ti)kj 0.5k

2

j

i 0.5(e2i + e

2

ij).

The first order conditions with respect to ei and eij are:

Uiei

= biki ei = 0,

Uieij

= bijkj eij = 0.

Ignoring the (NN) constraint for now, we have ei = biki/ and eij = bijkj. This is the (IC1)

constraint in the class of linear mechanisms.

The division managers choices of {ei, eij} are again independent of the true types ti. The (IC2)

constraint can be rewritten as:

ti arg max EUi(ti, ti)

= Wi(ti) + tiEtj [bi(ti, tj)ki(ti, tj) + bij(ti, tj)kj(ti, tj)].

where Wi(ti) represents the terms in manager is expected utility that only depend on her own report,

ti.

The Envelope Theorem implies

dEUi(ti, ti)

dti= Etj [bi(ti, tj)ki(ti, tj) + bij(ti, tj)kj(ti, tj)].

Integrating the above expression and imposing EUi(0) = U from the (IR) constraint yields:

EUi(ti) = U +

Zti0

Zt0

(biki + bijkj)dFjdsi. (8)

Integrating by parts over ti yields:

EUi = U + Zt

0 Zt

0

(biki + bijkj)idFidFj = U + Zt

0 Zt

0

(ei

+eij

)idFidFj

where the last equality follows from optimal choices ofei and eij by manager i.

Substituting Ewi = EUi + 0.5E(e2i + e

2ij) into E gives

E =

Zt0

Zt0

nnk1 + (e1 + e21)k1 + (t1 + t2)k1 0.5k

2

1 U1 0.5(e2

1+ e2

12)

+nk2 + (e2 + e12)k2 + (t2 + t1)k2 0.5k2

2 U2 0.5(e2

2+ e2

21)o

dF1dF2

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=

Zt0

Zt0

nE1V1 + E2V2 0.5(e

2

1+ e2

12) 0.5(e22 + e

2

21)

(

e1 +

e12 )1 (

e2 +

e21 )2

odF1dF2 2U .

Note that this is identical to the objective function in problem (P2). Therefore, point-wise differ-

entiation of the integrand will give identical first-order conditions as Equation 5, leading to the same

solution {ki, ei , e

ij} as Equation 6. To derive other parts of the optimal linear mechanism, we have

bi =eiki

= 1 i2ki

and bij =eijkj

= 1 i2kj

. The salary portion of the compensation can be

recovered from

ai = U +

Zti0

(biki + bijkj)ds + 0.5(e2

i + e2

ij )

bih

nki + (e

i + e

ji)ki + (ti + tj)ki 0.5k2i

ibij

hnkj + (e

j + e

ij)kj + (tj + ti)kj 0.5k2

j

i.

Since this optimal linear mechanism has the same solution {ki, ei , e

ij} as the optimal solution to problem

(P2), we have

EL = EP2,

where EL is the headquarters expected payoff in the optimal linear mechanism.

On the other hand, note that the optimal linear mechanism satisfies all the constraints in (P1)

hence it is a feasible mechanism of (P1) so it follows that

EL EP1.

Combining these two, we have EP2 EP1. But Step 1 establishes that EP2 EP1.

Therefore, EP2 = EP1, so the optimal linear mechanism is an optimal mechanism in Program (P1).

So far we have studied the Bayesian Nash implementation of the headquarters mechanism design

problem. Since the optimal mechansim is monotonic in both t1 and t2, by the results of Mookherjee

and Reichelstein (1992), it can also be implemented in dominant strategies. This completes the proof.

Q.E.D.

Proof of Corollary 1: The monotonicity of the capital allocations, managerial resource allocations,

and performance-pay are immediate after taking the derivative of each with respect to t1 and t2.We now prove that salary a1(t1, t2) is decreasing in t1 (a similar argument holds for a2(t1, t2) decreasing

in t2). Consider the two values, M1 and M2, given by:

M1 =

Zt10

(b1k1)ds1 b1h

nk1 + (e1 + e21)k1 + (t1 + t2)k1 0.5k2

1

i+ 0.5e21,

M2 =

Zt10

(b12k2)ds1 b12h

nk2 + (e2 + e12)k2 + (t2 + t1)k2 0.5k2

2

i+ 0.5e2

12.

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Since the sum of M1 and M2 is the equal to the salary (less a constant U), it suffices to show that M1

and M2 decrease with t1. Note that M1 can be written as

M1 =Z

t1

0

(b1k1)ds1 b1k1t1 0.5b1k1(n + e21 + t2 t1).

Thus,

M1t1

= 0.5(b1k1)

t1(n + e21 + t1 + t2) 0.5b1k1

e21t1

+ 0.5b1k1

< 0.5b1hk1t1

(n + e21 + t1 + t2) k1i

= 0.5b1(12

2

)1

he21 + 1 + 2 (n + e21 + t1 + t2)

d1dt1

i< 0

where the first inequality follows from the monotonicity ofb1 and e21, and the second inequality follows

from the monotonicity of 1 and Assumptions (A1) and (A2). Similarly we can show that M2 also

decreases with t1. Q.E.D.

Proof of Implication 1:

Using the solutions for the first-best capital allocation, kFBi , and the second-best capital allocation,

ki, from Propositions 1 and 2 we have:

kFB1 k1 = (1

2

2

)1(1 + 2) > 0 (9)

implying capital underinvestment in the second-best mechanism. Since

d1

dt1 < 0 and

d2

dt2 < 0, kFB

1 k1decreases with t1 and t2. The comparative statics are obvious after taking the partial derivatives with

respect to the parameters. Similarly, we can prove the properties of kFB2 k2.

Note also that:

eFB1 e1 =

(kFB

1 k1) + (

)1 > 0.

All the results for underinvestment in resources deployed in the managers own project, e1, follow

immediately. Similar arguments apply to e2, e12 and e21. Q.E.D.


Using the solutions for the first-best capital allocation, kFBi , and the second-best capital allocation,

ki, from Propositions 1 and 2 we have:

(ki kFBi )

tj=

(0j)

1 2/ 2/> 0

where the inequality results from the monotonicity of j and Assumption (A1). This proves part (i).

Part (ii) follows from taking the derivative of the right-hand side with respect to each parameter. Q.E.D.

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If k1 = k2 = k then the division-level performance pay is given by:

bi bij = i

2kj i

2ki= i

k( 2

2). (10)

The desired result is clear when (bi bij) > 0 and you substitute the second-best capital allocation

from Proposition 2 into the above.

Ifk1 = k2 = k then the relative division-level performance pay compared to firm-level performance

pay is given by:

bi bijbij

=bibij

1 =1 i

2k

1 i2k

1

Taking logs yields:

ln(1 i

2k

1 i2k

) = ln(1 i2k

) ln(1i2k

) i

k(

2

2)

where the approximation is valid if n is significantly large. Since this last expression is identical to the

above, the same comparative statics results hold. Q.E.D.


We need to show that bij and bji are positively correlated. For what follows, we use the fact:

Cov(p(x), q(x)) > 0 if p0(x), q0(x) > 0. Denote yij =i

n + (tj j) + (ti i). It suffices to show

that yij and yji are positively correlated since Corr(bij , bji) = Corr(yij , yji). Given ti, yij and yji

increase with tj . Using our fact above we have E(yijyji |ti) > E(yij|ti)E(yji |ti). Also, since E(yij|ti)

and E(yji |ti) increase with ti, Eh

E(yij |ti)E(yji |ti)i

> Eh

E(yij |ti)i

Eh

E(yji |ti)i

. Therefore,

E(yijyji) = Eh

E(yijyji|ti)i

> Eh

E(yij |ti)E(yji |ti)i

> Eh

E(yij|ti)i

Eh

E(yji |ti)i

= E(yij)E(yji),

implying yij and yji are positively correlated. Q.E.D.

Proof of Implication 6: For the first statement, note that:

k1 k1

1=

n + (t1 1) + (t2 2(t2))

1 2

2

n + (t1 1) + t2 + e21

1 2

which increases in t2

. If there exists tcrit2 (0, t) such that k1k

11

= 0, then clearly k1 is larger (smaller)

than k11

when t2 is larger (smaller) than tcrit2

.

For the second statement, note that E{t1,t2}[k1 k11

] increases in . If there exists crit > 0 such

that E{t1,t2}[k1k11] = 0 then E{t1,t2}[k1] is larger (smaller) than E{t1,t2}[k

11 ] when is larger (smaller)

than crit. Q.E.D.

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Tony Bernardom